group theory

# Contents

## Definition

For a prime number $p$, the Prüfer $p$-group is defined uniquely up to isomorphism as the group where every element has exactly $p$ $p^{th}$ roots. It is a divisible abelian group which can be described in several ways, for example:

As such, it is the initial algebra of the functor $F: \mathbb{Z} \darr LCHAb \to \mathbb{Z} \darr LCHAb$ that pushes out along multiplication by $p: \mathbb{Z} \to \mathbb{Z}$.

## Properties

The Prüfer $p$-groups are the only infinite groups whose subgroups are totally ordered by inclusion. They are often useful as counterexamples in algebra; for example, a Prüfer group is an Artinian but not a Noetherian $\mathbb{Z}$-module.

Revised on August 27, 2014 03:55:54 by David Corfield (129.12.18.145)